A rigidity theorem for Kolmogorov-type operators

Let $D\subseteq {\mathbb{R}}^n$, $n\ge 3$, be a bounded open set and let $x_0\in D$. Assume that the Newtonian potential of $D$ is proportional outside $D$ to the Newtonian potential of a mass concentrated at $\{x_0\}\mathrm{.}$ Then $D$ is a Euclidean ball centered at $x_0$. This Theorem, proved by Aharonov, Shiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki–Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.